Question: Cauchy-Riemann test of analyticity of a complex function ...

CauchyRiemann:=proc(expr::algebraic) # original procedure
  local x, y, u, v, u_x, u_y, v_x, v_y, flag1, flag2;

  u:=evalc(Re(eval(expr, z=x+I*y)));
  v:=evalc(Im(eval(expr, z=x+I*y)));

  u_x:=diff(u,x);
  u_y:=diff(u,y);
  v_x:=diff(v,x);
  v_y:=diff(v,y);

  print('f(z)'=expr);
  printf("\n");
  
  print('u(x,y)'=u);
  print('u[x](x,y)'=u_x);
  print('u[y](x,y)'=u_y);
  printf("\n");

  print('v(x,y)'=v);
  print('v[x](x,y)'=v_x);
  print('v[y](x,y)'=v_y);
  printf("\n");

  if u_x=v_y then
    print('u[x]=v[y]');
    print(u_x=v_y);
    flag1:=true;
  else
    print('u[x]<>v[y]');
    print(u_x<>v_y);
    flag1:=false;
  end if;

  if u_y=-v_x then
    print('u[y]=-v[x]');
    print(u_y=-v_x);
    flag2:=true;
  else
    print('u[y]<>-v[x]');
    print(u_y<>-v_x);
    flag2:=false;
  end if;
  
printf("\n");
if flag1=true and flag2=true then
   print(`Fullfill the Cauchy-Riemann Equations`);
   print(`The derivative is:`='u[x]+I*v[y]');
   print('diff(f(z),z)'=u_x+I*v_y);
else
   print(`Cauchy-Riemann ?`);
end if

end proc:

f(z):=1/(z+2):
CauchyRiemann(f(z))

restart;

# Improved and corrected version of the CauchyRiemann procedure :ASKED AI 
CauchyRiemann := proc(expr::algebraic)
    local x, y, u, v, u_x, u_y, v_x, v_y, CR1, CR2;

    # Assign real and imaginary parts of the function
    u := evalc(Re(eval(expr, z = x + I*y)));
    v := evalc(Im(eval(expr, z = x + I*y)));

    # Calculate partial derivatives
    u_x := diff(u, x);
    u_y := diff(u, y);
    v_x := diff(v, x);
    v_y := diff(v, y);

    # Properly format and print function details
    printf("f(z) = %a\n", expr);
    printf("u(x, y) = %a, u_x = %a, u_y = %a\n", u, u_x, u_y);
    printf("v(x, y) = %a, v_x = %a, v_y = %a\n", v, v_x, v_y);

    # Evaluate and print Cauchy-Riemann equations
    CR1 := u_x = v_y;
    CR2 := u_y = -v_x;
    printf("\nCauchy-Riemann Equations:\n");
    printf("u_x = v_y: %a\n", CR1);
    printf("u_y = -v_x: %a\n", CR2);

    # Check both equations
    if CR1 and CR2 then
        printf("The function is analytic (holomorphic) at this point.\n");
        printf("The derivative f'(z) is %a + I*%a\n", u_x, v_y);
    else
        printf("The function does not satisfy the Cauchy-Riemann equations and is not analytic.\n");
    end if;
end proc;

# Test the procedure with a specific function
f := z -> 1/(z + 2);
CauchyRiemann(f(z));

f(z) = 1/(z+2)
u(x, y) = (x+2)/(y^2+(x+2)^2), u_x = 1/(y^2+(x+2)^2)-(x+2)/(y^2+(x+2)^2)^2*(2*x+4), u_y = -2*(x+2)/(y^2+(x+2)^2)^2*y
v(x, y) = -y/(y^2+(x+2)^2), v_x = y/(y^2+(x+2)^2)^2*(2*x+4), v_y = -1/(y^2+(x+2)^2)+2*y^2/(y^2+(x+2)^2)^2

Cauchy-Riemann Equations:
u_x = v_y: 1/(y^2+(x+2)^2)-(x+2)/(y^2+(x+2)^2)^2*(2*x+4) = -1/(y^2+(x+2)^2)+2*y^2/(y^2+(x+2)^2)^2
u_y = -v_x: -2*(x+2)/(y^2+(x+2)^2)^2*y = -y/(y^2+(x+2)^2)^2*(2*x+4)
The function does not satisfy the Cauchy-Riemann equations and is not analytic.